Stability Analysis Of The Splitting Step θ Method For Two Classes Of Stochastic Delay Differential Equations

Stochastic differential equations(SDEs)are a class of equations which influ-enced by random factors based on the deterministic differential equations,and it has been widely used in biotechnology,genetics,physics,chemistry,economics,fi-nance as well as other fields.The stochastic delay differential equations(SDDEs)are not only related to the current state,but also related to some past states with noise interference.Due to it is difficult to obtain the true solution for most of SDDEs in real life,it is necessary to study the numerical methods for such systems.This paper mainly studies the stability of numerical methods for two kinds of SDDEs.The first one is devoted to the stability of split-step one-leg 0 method(SSOLTM)for SDDEs,and the corresponding numerical examples are given to illustrate the stability of the method;the second part is concerned with a class of neutral stochastic delay integral differential equations extended on SDDEs and a class of splitting steps 0 method(SST)to solve the equations.The main works of this article are as follows:In the first chapter,the current development is introduced on the stability of the split-step numerical method of SDDEs.Moreover,the advances on the study of the stochastic delay integral differential equations and the SSOLTM are introduced.Chapter 2 is concerned with SSOLTM for SDDEs.When the drift term coef-ficient satisfies non-globally Lipschitz,in the case 1/2≤θ≤1,the mean square asymptotically stability can be obtained for solving SDDEs;in the case 0≤θ<1/2,the stability can be proved as well if the drift term coefficient satisfies the linear growth condition further.Finally,the correctness of the theoretical results is veri-fied by numerical experiments.In the third chapter,mean square exponential stability of the exact solution for neutral stochastic delay integral differential equations is examined when the drift term coefficient satisfies the linear growth condition.SST is mean square exponential stable for stepsize h<h*as θ∈[0,1/2].While θ∈(1/2,1],we obtain the mean square exponential stability for any step size h=τ/m.Finally,the numerical experiment verified the effectiveness of the theoretical results.